For an integer-valued compound Poisson process with geometrically distributed jumps of a certain sign [these processes are called almost upper (lower) semicontinuous] defined on a finite regular Markov chain, we establish relations (without projections) for the moment-generating functions of extrema and their complements. Unlike the relations obtained earlier in terms of projections, the proposed new relations for the moment-generating functions are determined by the inversion of the perturbed matrix cumulant function. These matrix relations are expressed via the moment-generating functions for the distributions of the corresponding jumps.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 8, pp. 1034–1049, August, 2015.
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Herych, M.S., Husak, D.V. On the Moment-Generating Functions of Extrema and Their Complements for Almost Semicontinuous Integer-Valued Poisson Processes on Markov Chains. Ukr Math J 67, 1164–1182 (2016). https://doi.org/10.1007/s11253-016-1144-2
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DOI: https://doi.org/10.1007/s11253-016-1144-2